What difference does it make if the tails in the Double Fisherman's bend end up on opposite sides vs. the same side?
Depends on precisely what you mean by "difference".
Does it mean that the direction of the strangle turns w.r.t. each other does not change any of the bend's properties?
Depends on what you precisely mean by "properties".
Both of these questions appear to be of the same genesis - just formulated differently.
The following is a a longer answer (since this is a technical knot geek forum - a longer reply seems reasonable):
In the first instance, all knots have a particular geometric form which is unique and enables it to be distinguished from other knots. We recognise knots by their distinctiveness. And so this is one possible answer to your opening questions. That is, if you alter the geometry of a knot, you change if into something else.
Of course, the mirror inverted form of a knot is the same - that is, tie any knot and hold it up to a plane mirror - the reflection you see is the mirror is opposite, but still just as valid. Left handers will often tie a knot in mirror form to a right-hander.
If your question relates to other factors such as jam resistance, security, stability, etc... these likely haven't been fully tested using the scientific method in peer reviewed technical papers. Most users of #1415 Double Fishermans are happy to always tie the form as depicted by Ashley and other authors - where the each 'strangle' is pointing in
relative opposition (but has same chirality).
#1415 Double Fishermans bend is an inter-penetrating geometry - where each strangle compresses against the other. This makes it very secure, but also horribly vulnerable to jamming.
I always advise that 'bends' used in life critical applications have
identical length tails - which serve as visual ques to alert the user if something has changed. In flat webbing, #1412 Water knot/ Ring bend/Tape knot can work itself loose under cyclic loading conditions - and I know of one tragic accident where a person in the USA fell to his death on account of a web sling coming undone (I might have posted about previously it in 'reports'). In my view, there was no well entrenched doctrine for checking the knot and the tails - an imbalance of which might have alerted his to an anomaly... we'll never know for sure.
NOTE: The propensity of webbing material to loosen under cyclic loading conditions is highly variable - one driver being the type of webbing the knot (ie #1412 Water knot) is tied in. The stiffness and frictive properties of webbing is highly variable from one manufacturer to the next.
As for the jam resistance and/or stability of a 'discordant #1415, there isn't any peer reviewed data that I can point to. However, one could surmise that it is possible to see some variation in jam resistance - ie the threshold load where jamming begins... possibly due to less uniform keying in of all rope segments as they compress against each other. But this is pure speculation and is an open question.
As for MBS yield (strength) - this is irrelevant within climbing, rescue, abseiling, canyoning, applications. Strength is often the thought process which dominates most testers - which is unfortunate. Some testers are starting to explore things other than pure MBS yield - eg jamming threshold - but it is rare to find technical papers that do so.
Note that my remarks are in relation to synthetic kernmantel type rope (not vegetable fibre as would have been evident at the time of Ashley.
